The square chromatic number of the torus

The square of a graph G denoted by G 2 , is the graph with the same vertex set as G and edges linking pairs of vertices at distance at most 2 in G . The chromatic number of the square of the Cartesian product of two cycles was previously determined for some cases. In this paper, we determine the precise value of ¿ ( ( C m ¿ C n ) 2 ) for all the remaining cases. We show that for all ordered pairs ( m , n ) except for ( 7 , 11 ) we have ¿ ( ( C m ¿ C n ) 2 ) = ¿ | V ( ( C m ¿ C n ) 2 ) | α ( ( C m ¿ C n ) 2 ) ¿ , where α ( G ) denotes the independent number of G . This settles a conjecture of Sopena and Wu (2010). We also show that the smallest integer k such that ¿ ( ( C m ¿ C n ) 2 ) ¿ 6 for every m , n ¿ k is 10. This answers a question of Shao and Vesel (2013).